Problem: $f(x)=\dfrac{4}{1+5x^3}$ Find a power series for $f$. Choose 1 answer: Choose 1 answer: (Choice A) A $4-20x^3+100x^6+\ldots +4(-5)^n x ^{3n}+\ldots$ (Choice B) B $4+20x^3+100x^6+\ldots +4\cdot 5^n x ^{3n}+\ldots$ (Choice C) C $1-5x^3+20x^6+\ldots +(-5)^n x ^{3n}+\ldots$ (Choice D) D $4+20x^3-100x^6+\ldots +4(-5)^n x ^{3n}+\ldots$
Answer: This is a geometric series with first term $a\text{ }=\text{ }4$ and common ratio $r\text{ }=\text{ }-5x^3$. Therefore, the series is as follows. $4-20x^3+100x^6+\ldots +4(-5)^n x ^{3n}+\ldots $